Example 1 (Unit-wise OLS) 
Let the outcome \(y\)  be determined by the covariate \(x\)  as \(y_{it} = \theta _i x_{it} + u_{it}\) , \(\operatorname{\mathbb {E}}(x_{it}u_{it})=0\) , \(i=1,\dots , N, t=1, \dots , T\) . We estimate heterogeneous coefficients \(\theta _i\)  by unit-by-unit application of OLS: \(\vartheta _{i, T}= \left(T^{-1}\sum _{t=1}^T x_{it}^2 \right)^{-1}T^{-1}\sum _{t=1}^T x_{it}y_{it}\) . Define \(\varepsilon _{i,T} = \left(T^{-1}\sum _{t=1}^T x_{it}^2 \right)^{-1}T^{-1/2}\sum _{t=1}^T x_{it}u_{it} \)  to write \(\vartheta _{i, T}\)  as in eq. (REF ) for \(p=1/2\) . More generally, let \(x_{it}\)  be a vector of covariates in the model: \(y_{it}=\beta _i^{\prime }x_{it}+u_{it}\) . In this case the parameter of interest \(\theta _i\)  may be the first coordinate of \(\beta _i\)  or a more general smooth transformation of \(\beta _i\) , such as a long-run effect in a dynamic model [1]}.

